About Optimal Prefix Codes over Countably Infinite Alphabets: Probabilistic Intervals for the Codeword Lengths Assignment

2026-04-19

AI Summary

This study addresses the problem of optimal prefix code length assignment for discrete memoryless sources over countably infinite alphabets. By analyzing the interval containing the maximum symbol probability, the work establishes an exact correspondence between this probability and the optimal code lengths. The main contributions include the first construction of explicit probability intervals corresponding to any given positive integer code length \(k\), a concise criterion to identify probability distributions yielding specific code length patterns (e.g., \(l_i = i\)), and a proof that when the maximum symbol probability lies within such an interval, the associated optimal code length is uniquely determined. Compared to existing results on anti-uniform sources, this work provides clearer and more readily verifiable sufficient conditions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

For the discrete memoryless sources with a countably infinite alphabet, we prove that for any positive integer $k$, there exists a corresponding probability interval such that if the largest symbol probability $p_{1}$ falls in this interval, the optimal code length for the symbol equals $k$. Furthermore, for infinite sources, we provide a criterion to determine probability distributions whose optimal code length assignment follows the pattern $l^{best}_{i}=i$, for $i\ge 1$. Compared with the existing conclusion for anti-uniform sources, the proposed criterion requires less information for verification.

Problem

• optimal prefix codes
• countably infinite alphabets
• codeword lengths
• probability intervals
• discrete memoryless sources

Innovation

• optimal prefix codes
• countably infinite alphabets
• codeword length assignment
• probability intervals
• anti-uniform sources